38,585 research outputs found
Dynamical properties of profinite actions
We study profinite actions of residually finite groups in terms of weak
containment. We show that two strongly ergodic profinite actions of a group are
weakly equivalent if and only if they are isomorphic. This allows us to
construct continuum many pairwise weakly inequivalent free actions of a large
class of groups, including free groups and linear groups with property (T). We
also prove that for chains of subgroups of finite index, Lubotzky's property
() is inherited when taking the intersection with a fixed subgroup of
finite index. That this is not true for families of subgroups in general leads
to answering the question of Lubotzky and Zuk, whether for families of
subgroups, property () is inherited to the lattice of subgroups generated
by the family. On the other hand, we show that for families of normal subgroups
of finite index, the above intersection property does hold. In fact, one can
give explicite estimates on how the spectral gap changes when passing to the
intersection. Our results also have an interesting graph theoretical
consequence that does not use the language of groups. Namely, we show that an
expander covering tower of finite regular graphs is either bipartite or stays
bounded away from being bipartite in the normalized edge distance.Comment: Corrections made based on the referee's comment
-Colored Graphs - a Review of Sundry Properties
We review the combinatorial, topological, algebraic and metric properties
supported by -colored graphs, with a focus on those that are pertinent
to the study of tensor model theories. We show how to extract a limiting
continuum metric space from this set of graphs and detail properties of this
limit through the calculation of exponents at criticality
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
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